U.U.D.M. Project Report 2013:18 René Descartes’ Foundations of Analytic Geometry and Classification of Curves Sofia Neovius Examensarbete i matematik, 15 hp Handledare och examinator: Gunnar Berg Juni 2013 Department of Mathematics Uppsala University Abstract Descartes' La Géométrie of 1637 laid the foundation for analytic geometry with all its applications. This essay investigates whether the classification of curves presented in La Géométrie, into geometrical and mechanical curves based on their construction as well as into classes based on their equations, limited the further development of analytic geometry as a field. It also looks into why Descartes' further classification was algebraic rather than geometrical; and how it was criticized and why. In order to answer these questions, the essay touches on the historical background to Descartes' works, provides an overview and analysis of the ideas put forward by Descartes, and describes the development of analytic geometry in the 150 years following the publication of La Géométrie. Content 1. Introduction .................................................................................................................................................. p. 3 2. Mathematical context i.The Mathematical Background to Descartes' works …........................................................................ p. 4 ii. Overview of the content of La Géometrie …..................................................................................... p. 6 3. The classification of curves i. The classification itself …...................................................................................................................... p. 14 ii. The need for a classification …............................................................................................................ p. 16 4. Progress made after 1637 i. The development of analytic geometry ….......................................................................................... p. 20 ii. The role of curve construction …....................................................................................................... p. 22 5. Conclusion ….............................................................................................................................................. p. 25 6. References …................................................................................................................................................ p.28 1 1. Introduction Most of our modern mathematics is possible due to the use of functions and curves that can be visualized in a coordinate plane. The concept of a function and of curves as described by a relationship between two or more variables, is a relatively recent invention in mathematical history. In 1637, René Descartes (1596-1650), a French mercenary, mathematician and philosopher, published a book called La Géométrie as an appendix to his great Discours de la Methode pour bien conduire la raison, et chercher la verité dans les sciences. In La Géométrie, he set out to create a new, all-encompassing field of mathematics, where the then separate fields of ”true mathematics”, geometry and algebra, were linked together and used in symbiosis1. It was groundbreaking in the sense that it provided the entire mathematical community with a new set of tools: a way of solving algebraic equations using geometry; and a way of describing geometrical problems in algebraic terms, thus making it possible to manipulate and solve them2. First presented in 1637, these tools were developed and scrutinized throughout the century until the new discoveries eventually led to the creation of the calculus in 1666 and 16843. Today, calculus and coordinate geometry have an immense number of applications and are used in as separate fields as airplane coordination, astrophysics, farming and engineering. But despite its great impact, everything postulated in La Géométrie was far from indisputable. One must note that independently and at roughly the same time, Pierre de Fermat presented his own version of analytic geometry, stating in a short treatise that ”[w]henever in a final equation two unknown quantities are found, we have a locus, the extremity of one of these describing a line, straight or curved”4. Due to less exposure (it was only present in manuscript form until 1679) and a less modern algebra, Fermat's theories did not get as much exposure as La Géométrie, leaving Descartes to set the main foundation of analytic geometry as a field of study. In contrast to Fermat's, Descartes' algebra was much like the one used today. He was the first to systematically use letters near the end of the alphabet to represent unknowns; to use letters from the beginning of the alphabet to represent parameters or coefficients; and to denote exponents as an etc. The only exception was a2, which was written as aa to avoid typographical errors5. Descartes' symbolic algebra differed from that of, for example, François Viète, in that he saw all quantities of a polynomial expression as one-dimensional lines, thus eliminating the need to keep the homogeneity of an expression6. In the expression x3+ax+c, a would traditionally be interpreted as an area and c as a volume in order to keep the whole expression in the same dimension. Descartes argued that, because of the ratio 1 : x = x : x2 = x2 : x3, expressions such as x2y3 - y could be considered without any inherent contradictions since terms of higher powers (dimensions) can always be divided by the unit (1) to make them of a lower power (dimension)7. This step away from the classical idea of homogenous, and more reality-bound expressions, simplified working with terms of different powers and made it possible to use equations of higher degrees than three for geometrical problem solving. The notation in La Géometrie is very similar to our modern notation, and was a prerequisite for the simplicity of the concepts and methods Descartes presented8. During the development of his new mathematical structure, Descartes opened it up for a vast array of new curves to be used in geometrical problem solving. The need then arose to classify these new curves as geometrically acceptable or unacceptable. The traditional geometric curves had been known since Antiquity and could be constructed using straight-edge rulers and compasses; the new geometric curves (today called algebraic curves) were according to Descartes all those that could be constructed by the 1 Sasaki, 2003, p. 3 2 Sjöberg, 1996, p. 112f 3 Lund, 2002, p. 48: By Sir Isaac Newton and Gottfried Wilhelm Leibniz respectively. 4 Boyer, 2004, p. 75 5 Suzuki, 2002, p. 347 6 Sjöberg, 1996, p. 115 7 op.cit, pp. 114-115 8 Sasaki, 2003, p. 107 2 intersection of existing curves or could be traced using continuous motions of a known relation, thus fulfilling the criterion of geometric exactness. The third category consisted of the mechanical curves, which could not be constructed in this manner9. In order to make his method of problem solving mathematically legitimate, Descartes could only consider the geometric curves acceptable for it. A large part of La Géométrie is therefore devoted to the exact construction and further classification of these curves. Descartes' basic distinction between ”geometric” and ”mechanical” curves was based on earlier works dating as far back as 300 BC, specifically Euclid's Elements and Pappus' Collection (approximately 300 AD), as well as on works by the foremost geometer in the beginning of the 17th century, Christopher Clavius10. His further classification of geometric curves into classes, based on their algebraic equations, was however entirely new and has been much discussed. This essay aims to investigate what limitations Descartes created for his new field of mathematics because of this classification of curves, how it has been criticized and why, and how the development following Descartes´ works might have changed had all curves been deemed acceptable for his new method of problem solving. This will be done through an examination of the mathematical background to Descartes' works; an overview of the three books of La Géométrie; a section concerning his classification of curves and the need for this; a section on what results in analytic geometry came from mathematicians following Descartes; and a discussion on whether or not those developments could have been different or made quicker had all curves been accepted for problem solving. The quotations from La Géométrie are translated from the 1886 edition published by Hermann for Librairie Scientifique, with some inspiration from the translation by Smith and Latham11 and that found in Bos' On the Representation of Curves in Descartes' Géométrie12. 2. Mathematical context 2i. The Mathematical Background to Descartes' Works René Descartes was born in 1596 at La Haye, the son of a wealthy family. Of a frail constitution, the young Descartes was allowed to stay in bed until late in the morning, time that he used for contemplation and meditation and that is thought to have become “the source of the most important philosophical results that his mind produced”13. After receiving a law degree at the University of Poitiers in 1616, Descartes spent a year in France before travelling to Holland and enlisting at the military school in Breda14. During this time, in October 1618, he met with Isaac Beeckman, who would influence and inspire him to engage in the study of natural philosophy through mathematics. As Sasaki states: “in 1619 Descartes began to confess